A parametric curve will be continuous if all of its component functions are continuous.

For parametric curves in 2-space, this means that the curve will be continuous if the following condition is met:

For some arbitrarily small εx and εy and for every value t0 of t, it is always possible to choose a &delta greater than 0 but small enough that for the part of the curve such that t0 - &delta ≤ t ≤ t0 + &delta,

x(t0) - εx ≤ x(t) ≤ x(t0) + εx and

y(t0) - εy ≤ y(t) ≤ y(t0) + εy.

A parametric curve in 3-space will be continuous if the following condition is met:

For some arbitrarily small εx, εy, and εz and for every value t0 of t, it is always possible to choose a &delta greater than 0 but small enough that for the part of the curve such that t0 - &delta ≤ t ≤ t0 + &delta,

x(t0) - εx ≤ x(t) ≤ x(t0) + εx,

y(t0) - εy ≤ y(t) ≤ y(t0) + εy, and

z(t0) - εz ≤ z(t) ≤ z(t0) + εz.

Demos

Continuity of Parametric Curves in 2-Space

See if for arbitrary small epsilonX and epsilonY, it is possible to choose a nonzero delta small enough that the magenta part of curve lies inside the yellow rectangular box for certain values t0 of t, particularly those where there appears to be a discontinuity (if there are such values). If the x-component of the parametric curve is continuous, it will be possible to get the magenta part to lie between the red lines given a small enough delta. If the y-component is continous, it will be possible to get the magenta part to lie between the blue lines. If both parts are continous, the parametric curve will be continous and it will be possible to fit the magenta part of the curve inside the box.

Note for extremely small numbers (on the order of 10-16) the application will round to 0.

Continuity of Parametric Curves in 3-Space

This demo resembles the one above, with the principal difference here being the presence of an additional coordinate (z). See if, given an arbitrarily small box and some value t0 of t, it is possible to find a nonzero delta small enough to fit the magenta part of the curve inside the box.